Fermat's spiral

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedian and hyperbolic spiral. It divides the plane (like a line or circle or parabola) into two connected regions. But this division is less obvious than the division by a line or circle or parabola. It is not obvious to which side a chosen point belongs.

Definition of sector (light blue) and polar slope angle ${\displaystyle \alpha }$

From vector calculus in polar coordinates one gets the formula

${\displaystyle \tan \alpha ={\frac {r'}{r}}}$

for the polar slope and its angle ${\displaystyle \alpha }$ between the tangent of a curve and the corresponding polar circle (s. diagram).

For Fermat's spiral ${\displaystyle r=a{\sqrt {\varphi }}}$ one gets

• ${\displaystyle \tan \alpha ={\frac {1}{2\varphi }}.}$

Hence the slope angle is monotonely decreasing.

From the formula

${\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}}$

for the curvature of a curve with polar equation ${\displaystyle \;r=r(\varphi )\;}$ and its derivatives ${\displaystyle \;r'={\tfrac {a}{2{\sqrt {\varphi }}}}={\tfrac {a^{2}}{2r}}\;}$ and ${\displaystyle \;r''=-{\tfrac {a}{4{\sqrt {\varphi }}^{3}}}=-{\tfrac {a^{4}}{4r^{3}}}\;}$ one gets the curvature of a Fermat's spiral:

• ${\displaystyle \kappa (r)={\frac {2r(4r^{4}+3a^{4})}{(4r^{4}+a^{2})^{3/2}}}.}$

At the origin the curvature is ${\displaystyle 0}$. Hence the complete curve has

• at the origin an inflection point and the x-axis is its tangent there.

The area of a sector of Fermat's spiral between two points ${\displaystyle \;(r(\varphi _{1}),\varphi _{1})\;}$ and ${\displaystyle \;(r(\varphi _{2}),\varphi _{2})\;}$ is

${\displaystyle {\underline {A}}={\tfrac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi ={\tfrac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi \;d\varphi ={\frac {a^{2}}{4}}{\big (}\varphi _{2}^{2}-\varphi _{1}^{2}{\big )}}$

${\displaystyle \quad ={\frac {a^{2}}{4}}{\big (}\varphi _{2}+\varphi _{1}{\big )}{\big (}\varphi _{2}-\varphi _{1}{\big )}\;.}$

Fermat's spiral:
area between neighbored arcs

After raising both angles by ${\displaystyle 2\pi }$ one gets

${\displaystyle {\overline {A}}={\frac {a^{2}}{4}}{\big (}\varphi _{2}+\varphi _{1}+4\pi {\big )}{\big (}\varphi _{2}-\varphi _{1}{\big )}={\underline {A}}+a^{2}\pi {\big (}\varphi _{2}-\varphi _{1}{\big )}\;.}$

Hence the area ${\displaystyle A}$ of the region between two neighboring arcs is

• ${\displaystyle A=a^{2}\pi {\big (}\varphi _{2}-\varphi _{1}{\big )}\;.}$

${\displaystyle A}$ only depends on the difference of the two angles, not on the angles themselves.

For the example shown in the diagram, all neighboring stripes have the same area: ${\displaystyle A_{1}=A_{2}=A_{3}}$.

This property is used in electrical engineering for the construction of variable capacitors. [2]

the regions in between (white, blue, yellow) have all the same area, which is equal to the area of the drawn circle.

Special case due to Fermat

In 1636, Fermat wrote a letter [3] to Marin Mersenne which contains the following special case:

Let ${\displaystyle \varphi _{1}=0,\varphi _{2}=2\pi }$, then the area of the black region (see diagram) is ${\displaystyle A_{0}=a^{2}\pi ^{2}=}$ half of the area of the circle ${\displaystyle K_{0}}$ with radius ${\displaystyle r(2\pi )}$. The regions between neighboring curves (white, blue, yellow) have the same area ${\displaystyle A={\color {red}2}a^{2}\pi ^{2}}$ Hence:

• The area between two arcs of the spiral after a full turn equals the area of the circle ${\displaystyle K_{0}}$.

The length of the arc of Fermat's spiral between two points ${\displaystyle \;(r(\varphi _{1}),\varphi _{1}),(r(\varphi _{2}),\varphi _{2})\;}$ can be calculated by the integral:

${\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi =\cdots ={\frac {a}{2}}\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {1}{\varphi }}+4\varphi }}\,\mathrm {d} \varphi \;.}$

This integral leads to an elliptical integral, which can be solved numerically.

The inversion of Fermat's spiral (green) is a lituus spiral (blue)

The inversion at the unit circle has in polar coordinates the simple description: ${\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }$.

• The image of Fermat's spiral ${\displaystyle \ r=a{\sqrt {\varphi }}\ }$ under the inversion at the unit circle is a Lituus spiral with polar equation ${\displaystyle \;r={\frac {1}{a{\sqrt {\varphi }}}}\;}$.

For ${\displaystyle \;\varphi ={\tfrac {1}{a^{2}}}\;}$ both curves intersect at a fixpoint on the unit circle.

• The tangent (${\displaystyle x}$-axis) at the inflection point (origin) of Fermat's spiral is mapped onto itself and is the asymptotic line of the lituus spiral.